Andrew R. Kustin

Poincaré series of modules over local rings of small embedding codepth or small linking number

Abstract Let (S, n, k ) be a commutative noetherian local ring and M be a finitely generated S -module. Then the Poincare series of M is rational if any of the following holds: edim S -depth S ⩽ 3; edim S —depth S = 4 and S is Gorenstein; S is one link from a complete intersection; S is two links from a complete intersection and S is Gorenstein. To prove this result we assume, without loss of generality, that S = R I with ( R , m) regular local and ⊆. The key to the argument is to produce a factorization of the canonical map R → S as a composition of a complete intersection R → C with a Golod map C → S . This is accomplished by invoking a theorem of Avramov and Backelin once one has existence of a DGΓ-algebra structure on a minimal R -free resolution of S together with detailed knowledge of the structure of the induced homology algebra Tor R ( S , k ) = H ( K S ). Linkage theory provides the main technical tool for this analysis.

Structure theory for a class of grade four Gorenstein ideals

AssTRAcr. An ideal I in a commutative noetherian ring R is a Gorenstein ideal of grade g if pdR(R/I) = grade I = g and the canonical module ExtR(R/I, R) is cyclic. Serre showed that if g = 2 then I is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case g = 3. We present generic resolutions for a class of Gorenstein ideals of grade 4, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of grade 4 in k[[x, y, z, v]] that are n-generated for any odd integer n > 7. We construct other examples from almost complete intersections of grade 3 and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of grade 4, and which may be the key to a complete structure theorem.

Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four

Recently Buchsbaum and Eisenbud [3] exploited the algebra structure on a finite free resolution of a Gorenstein ideal of codimension three to obtain a complete determinantal description of such an ideal. As they pointed out, the study of algebra structures on resolutions has for the most part been confined to the (generally infinite) minimal free resolution of the residue field of a local ring or the Koszul resolution of an ideal generated by a regular sequence. They proposed, however, to extend the scope of the study to all minimal free resolutions of cyclic modules. Khinich [1] furnished an example of a grade four ideal I for which the minimal resolution of R/I does not admit the structure of an associative, differential, graded commutat ive algebra (DGC algebra). Khinichs ring R/I is Cohen-Macaulay, but not Gorenstein. We conjecture that minimal finite free resolutions of Gorenstein factor rings R/a admit D G C algebra structures. In this paper we establish the conjecture for R a Gorenstein local ring in which 2 is a unit and a a Gorenstein ideal of grade (or height) four. We begin by clarifying what we mean by an algebra structure on a resolution. Let

The resolution of the generic residual intersection of a complete intersection

The concept of residual intersection, introduced by Artin and Nagata [l] in 1972, is a fruitful generalization of linkage as the following two examples attest. Let I be a strongly Cohen-Macaulay ideal in a CohenMacaulay local ring R. If Z satisfies the condition (G,), then Huneke [6, Proposition 4.31 has proved that the extended Rees algebra R[It, t‘1 is defined by an ideal which is obtained from I by way of residual intersection. Since the extended Rees algebra is a deformation of both the Rees algebra R[It] and the associated graded algebra gZ’/I” ‘, it contains considerable information about the analytic properties of I. (The definitions of residual intersection, strongly Cohen-Macaulay, and (G,) may be found in Section 4.) Huneke [6, Theorem 4.11 has also shown that the ideal .I, generated by the maximal order minors of a generic n x m matrix, is a residual intersection of a generic codimension two Cohen-Macaulay ideal. Since J is rather poorly behaved with respect to linkage [S], it is promising that it is close to a well understood ideal once we weaken the tie from linkage to residual intersection.

The minimal free resolutions of the Huneke—Ulrich deviation two gorenstein ideals

Structure theorems exist for Gorenstein ideals of small grade. A different parameter one can use for organizing ideals is deviation (or complete intersection defect). The deviation of the ideal I is the minimal number of generators of I minus the grade of I. A deviation zero ideal is a complete intersection. Kunz [S] proved that there are no deviation one Gorenstein ideals. Presently people are trying to find all deviation two Gorenstein ideals. See, for example, [3]. A hypersurface section of a grade g 1 deviation two Gorenstein ideal is a grade g deviation two Gorenstein ideal. We consider this to be a trivial example. Very few non-trivial examples of deviation two Gorenstein ideals are known. The Buchsbaum-Eisenbud structure theorem [2] describes all grade three Gorenstein ideals. In particular, every deviation two grade three Gorenstein ideal is generated by the maximal order pfaflians of a 5 x 5 alternating matrix. No non-trivial deviation two Gorenstein ideals of even grade are known. Herzog and Miller [3] have taken a step toward proving that all grade four deviation two Gorenstein ideals are trivial. Let I be a grade four deviation two Gorenstein ideal. If I is a generic complete intersection and I/I* is CohenMacaulay, then I is a hypersurface section. The only known non-trivial deviation 2 Gorenstein ideals of grade at least four are the ideals of Huneke and Ulrich [4], which are the ideals studied in this paper. For each integer s, let y’“’ be a 1 x s matrix of indeterminates, X’“’ be an s x s alternating matrix of indeterminates, g’“’ be the product y(“X(‘), and ct.‘) be the sequence gp),..., g’,“! , . Let A, be the ideal of

Rational normal scrolls and the defining equations of Rees algebras

Abstract Consider a height two ideal, I, which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k[x, y]. Suppose that one column in the homogeneous presenting matrix φ of I has entries of degree n and all of the other entries of φ are linear. We identify an explicit generating set for the ideal which defines the Rees algebra ℛ = R[It]; so for the polynomial ring S = R[T 1, . . . , Tm ]. We resolve ℛ as an S-module and Is as an R-module, for all powers s. The proof uses the homogeneous coordinate ring, A = S/H, of a rational normal scroll, with . The ideal is isomorphic to the n th symbolic power of a height one prime ideal K of A. The ideal K (n) is generated by monomials. Whenever possible, we study A/K (n) in place of because the generators of K (n) are much less complicated then the generators of . We obtain a filtration of K (n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon–Northcott complexes. The generators of I parameterize an algebraic curve in projective m – 1 space. The defining equations of the special fiber ring ℛ/(x, y)ℛ yield a solution of the implicitization problem for .

A general resolution for grade four Gorenstein ideals

Gorenstein rings occur in a multitude of different guises: as rings of invariants, as coordinate rings of certain determinantal varieties and symmetric semigroup curves, as representatives of linkage classes, and so on. In an attempt to unify this motley collection of examples (at least for finite projective dimension) one seeks a generic free resolution whose specializations yield all examples of given embedding codimension. The present paper describes a resolution for codimension four, not generic, but general enough to encompass many diverse examples. The structure of this resolution is intimately related to the differential, graded, commutative algebra that it supports, and to the deformation theory of codimension four Gorenstein algebras. These ideas are brought together in the determination of the singular locus of certain codimension four Gorenstein varieties. More generally they suggest a classification of codimension four Gorenstein rings that begins to impose some order on the examples.

Linkage theory for algebras with pure resolutions

Abstract Let S be a graded Cohen-Macaulay quotient R I of a polynomial ring R = k [ X 1 ,…, X n ] over an infinite field k . If S is in the linkage class of a complete intersection then its Koszul homology and twisted conormal module are Cohen-Macaulay. By passing to zero- and one-dimensional specializations of S , one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection. We prove that if S is generically a complete intersection and the minimal free solution of S is pure and almost linear, then S is not in the linkage class of a complete intersection (apart from a few obvious exceptions). For algebras of small codimension the results are stronger: if S has codimension three and has a pure resolution, then S is in the linkage class of a complete intersection if and only if S is Gorenstein; if S is a codimension four Gorenstein algebra with a pure resolution then S is in the linkage class of a complete intersection if and only if S is a complete intersection.

Tight double linkage of Gorenstein algebras

Abstract A Gorenstein ideal K in a local ring R is in the class ℋ if there is a sequence of linked ideals I 0 ∼ I 1 ∼ ⋯ ∼ I 2 n = K with I 0 a complete intersection and I 2 i , Gorenstein for all i . If K is in ℋ, then K is linked to a complete intersection by a sequence of “tight double links.” This means there is a sequence I 0 = J 0 ∼ J 1 ∼ … ∼ J 2m = K , such that for each p , J 2 p + 1 is an almost complete intersection of the form ( b , y, w ) and its immediate neighbors are linked to it by the “similar” regular sequences ( b , y ) and ( b , w ). respectively. In this sense J 2 p and J 2 p + 2 are “tightly linked” Gorenstein ideals. Let k be an algebraically closed field and A = k 〚 X 1 ,…, X q 〛/ k with K in ℋ. The k -algebra A is rigid if and only if it is linked to a regular complete intersection by a sequence of semi-generic tight double links. It follows that if A is rigid, then it is regular in codimension six.

The bi-graded structure of symmetric algebras with applications to Rees rings ☆

Abstract Consider a rational projective plane curve C parameterized by three homogeneous forms of the same degree in the polynomial ring R = k [ x , y ] over a field k. The ideal I generated by these forms is presented by a homogeneous 3 × 2 matrix φ with column degrees d 1 ≤ d 2 . The Rees algebra R = R [ I t ] of I is the bi-homogeneous coordinate ring of the graph of the parameterization of C ; and accordingly, there is a dictionary that translates between the singularities of C and algebraic properties of the ring R and its defining ideal. Finding the defining equations of Rees rings is a classical problem in elimination theory that amounts to determining the kernel A of the natural map from the symmetric algebra Sym ( I ) onto R . The ideal A ≥ d 2 − 1 , which is an approximation of A , can be obtained using linkage. We exploit the bi-graded structure of Sym ( I ) in order to describe the structure of an improved approximation A ≥ d 1 − 1 when d 1 d 2 and φ has a generalized zero in its first column. (The latter condition is equivalent to assuming that C has a singularity of multiplicity d 2 .) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2 = d 1 d 2 and φ has a generalized zero in its first column, then we record explicit generators for A . When d 1 = d 2 , we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for A d 1 − 2 and the number of singularities of multiplicity d 1 that are on or infinitely near C . We conclude with a table that translates between the bi-degrees of a bi-homogeneous minimal generating set for A and the configuration of singularities of C when the curve C has degree six.